Fully developed flow in a long triangular channel under an applied magnetic field

Full closed form solutions are provided for the fluid flow impacted by a perpendicularly applied uniform magnetic field inside a pipe of triangular cross section. The governing equation of pressure gradient induced flow under the external magnetic field is reduced to the Helmholtz partial differential equation with Dirichlet boundary conditions on the scaled side lines of the equilateral triangle. The velocity solution is then derived in terms of elementary exponential functions involving the magnetic strength parameter, or the Hartmann number, controlling the retarding effect of magnetic field on the velocity profiles. The Lorentz force effects are clarified on the two-variable velocity variations as well as on the centerline velocity, volumetric flow rate and wall shears. The control of flow field and recirculating zone is visualized graphically increasing the strength of magnetic field. At the Hartmann number 625, it is anticipated that the velocity is nearly flattened at the vertical positions. The provided exact solutions are beneficial to the research community studying the triangular shaped pipes under different physical scenarios. The presented mathematical foundation can be used in recently popular nanotechnology facilitated by miniature channels.